LYAPUNOV EXPONENTS FOR Mob(D)-COCYCLES:
A PROOF OF OSELEDETS THEOREM IN DIMENSION 2
JAIRO BOCHI
Oseledets multiplicative ergodic theorem
Let T : M → M be an invertible measurable transformation, and µ be a T -invariant
probability on M . Let π : E → M be a finite-dimensional real vector bundle over M ,
endowed with a measurable Riemannian metric k · k. Let F : E → E be a measurable
vector bundle automorphism over T : this means that π ◦ F = T ◦ π and the action
Fx : Ex → ET (x) of F on each fiber Ex = π −1 (x) is a linear isomorphism. We also call F
a cocycle over T .
Example. (dynamical cocycle) Let T : M → M be a C 1 diffeomorphism on a Riemannian
manifold M , E be the tangent bundle of M , and F = DT be the derivative of T .
Example. (trivial bundle) Suppose E = M × Rd . Then each Fx is a d × d matrix, that
is, the cocycle corresponds to a map from M to the linear group GL(d, R).
Oseledets ergodic theorem states that, under an integrability condition, almost every
fiber splits as a direct sum of subspaces such that the iterates Fxn : Ex → ET n (x) have
well-defined rates of exponential growth, in norm, restricted to each subspace:
Theorem 1 (Oseledets [Ose68]). Assume log+ kFx k and log+ kFx−1 k are µ-integrable
(where log+ = max{0, log}). Then for µ-almost every x ∈ M , there exist k(x) ∈ N, real
k(x)
such that for every
numbers λ1 (x) > · · · > λk(x) , and a splitting Ex = Ex1 ⊕ · · · ⊕ Ex
1 ≤ i ≤ k(x) we have Fx (Exi ) = ETi (x) and
lim
n→±∞
1
log kFxn (vi )k = λi (x)
n
for all non-zero vi ∈ Exi .
Moreover, the Lyapunov exponents λi (x), and the Oseledets subspaces Exi are unique µalmost everywhere, and they depend measurably on x.
We are going to prove Oseledets theorem in the case when the vector bundle is 2dimensional. For simplicity we also suppose that it is a trivial bundle, that is, E = M ×R2 .
Then each Fx is given by a 2 × 2 matrix. We take these matrices to be in SL(2, R), that
is, to have determinant 1. This is not a restriction because the validity of the theorem
is not affected if we multiply F by some non-zero function θ (as long as the integrability
condition is preserved) : the Oseledets subspaces remain the same, and one adds the
Birkhoff average of log |θ| to the Lyapunov exponents.
There are several proofs of Oseledets theorem in the literature, besides the original
one. See for instance [Mañ87, Chapter 4]. Another proof of the 2-dimensional case can
be found in [You95].
Date:
1
2
JAIRO BOCHI
Angles between Oseledets directions. As a complement to theorem 1, we also prove
k(x)
that the angles between the spaces of the Oseledets splitting Ex = Ex1 ⊕ · · · ⊕ Ex
are
sub-exponential, in the sense made precise by the following:
k(x)
Proposition 2. Let Ex = Ex1 ⊕ · · · ⊕ Ex
be the splitting given by Oseledets theorem.
Fix an integer k with 2 ≤ k ≤ dim E and let J1 t J2 be a partition of the set of indices
bxi = ⊕j∈J Exj for i = 1,
{1, . . . , k} into two disjoint non-empty sets. If k(x) = k, denote E
i
2. Then, for µ-a.e. x satisfying k(x) = k, we have
1
b2 n
b1 n , E
log sin ^ E
lim
T (x) = 0.
T (x)
n→±∞ n
We will assume Oseledets theorem and deduce proposition 2 from two simple results,
proposition 3, from ergodic theory, and proposition 4, from linear algebra.
Proposition 3. Let B ⊂ M be a T -invariant positive measure set and ϕ : B → R be a
measurable function such that ϕ ◦ T − ϕ is integrable 1. Then limn→∞ ϕ(T n (x))/n = 0
for a.e. x ∈ B.
Proof. Let ψ = ϕ ◦ T − ϕ and let ψ̂ be the limit of the Birkhoff averages of ψ. Then
(ϕ ◦ T n )/n → ψ̂ a.e. in B. In particular, ψ̂(x) 6= 0 implies |ϕ(T n (x))| → ∞. But, by the
Poincaré’s recurrence theorem, the set of points x ∈ B which satisfy the latter condition
has zero measure. Therefore ψ̂ = 0 a.e. in B.
Proposition 4. Let L : Rd → Rd be a invertible linear tranformation and let v, w be
non-zero vectors. Then
1
sin ^(Lv, Lw)
≤
≤ kLk kL−1 k.
kLk kL−1 k
sin ^(v, w)
Proof. Recall that for any α ∈ R, kw + αvk ≥ kwk sin ^(v, w), with equality when α =
hv, wi/kvk2 .
Let β = hLv, Lwi/kLvk2 and z = w + βv. Then kzk ≥ kwk sin ^(v, w) and, on the
other hand, kLzk = kLwk sin ^(Lv, Lw). Therefore
sin ^(Lv, Lw) =
kLzk
kL−1 k−1 · kzk
sin ^(v, w)
≥
≥
,
kLwk
kLk · kwk
kLk kL−1 k
proving the first inequality. The second one is analogous.
Proof of proposition 2. Let B ⊂ M be the T -invariant set of points x where k(x) = k. The
b1 , E
b 2 ) is measurable. By proposition 4,
function ϕ : B → R defined by ϕ(x) = log sin ^(E
x
x
−1
we have |ϕ(T x) − ϕ(x)| ≤ log kFx k + log kFx k, which, by the hypothesis of the Oseledets
theorem, is integrable. So proposition 3 gives ϕ(T n (x))/n → 0 for a.e. x ∈ B.
Automorphisms of the disk
Mob(D) is the set of all automorphisms of the unit disk D = {z ∈ C; |z| < 1}, that
is, all conformal diffeomorphisms f : D → D. The canonical form of an automorphism
f ∈ Mob(D) is
f (z) = βϕα (z),
where
ϕα (z) =
z−α
, α ∈ D, |β| = 1.
1 − αz
1With the stronger assumption “ϕ is integrable” the proposition would be a direct consequence of
Birkhoff’s theorem.
OSELEDETS
3
The hyperbolic metric on the disk is given by
dρ =
2 |dz|
2
1 − |z|
.
Since the straight lines through the origin are geodesics, we have
Z |z|
dr
1 + |z|
(1)
ρ(z, 0) = 2
= log
= 2 arctgh |z| .
2
1
−
r
1
− |z|
0
Using the fact that the hyperbolic metric is invariant under every automorphism, we may
deduce its general expression
(2)
ρ(z1 , z2 ) = ρ(ϕz2 (z1 ), ϕz2 (z2 )) = ρ
z1 − z2
|z1 − z2 |
, 0 = 2 arctgh
.
1 − z1 z 2
|1 − z1 z 2 |
The proof of the following formula may be found in [Nic89, page 12].
Proposition 5. Let f ∈ Mob(D) and ξ ∈ ∂D. Then
2
|f 0 (ξ)| =
1 − |f (0)|
2
|ξ − f −1 (0)|
.
We denote by Mob(H) the set of automorphisms of the complex half-plane H = {z ∈
C; Im z > 0}. Each f ∈ Mob(H) has the form
az + b
, a, b, c, d ∈ R, ad − bc 6= 0.
cz + d
We also fix a conformal equivalence h : H → D, for instance,
f (z) =
(3)
h(z) =
i−z
.
i+z
Automorphisms versus matrices
There is a natural group isomorphism between Mob(H) and the projective linear group
PSL(2, R) = PGL(2, R), as follows. If
az + b
a b
∈ Mob(H) and A =
∈ SL(2, R)
f (z) =
c d
cz + d
then f |∂H = f |R∪{∞} describes the action of A on the projective space RP1 : the matrix
A maps the line with co-slope x (i.e. the line containing the vector (x, 1) ) to the line
with co-slope (ax + b)/(cx + d).
Remark. Identifying ∂H with RP1 via co-slopes in this way, we see that the map h :
H → D induces a homeomorphism φ : RP1 → ∂D defined by [(cos θ, sin θ)] 7→ e−2iθ .
Using the conformal equivalence h : H → D we find a corresponding isomorphism
between Mob(D) and PSL(2, R). Denote by fA ∈ Mob(D) the automorphism associated
to the projective class [A] = {±A} of each A ∈ SL(2, R). We leave it to the reader to
check that fA (0) = 0 if and only if A is a rotation.
Remark. Corresponding to the canonical form f (z) = βϕα (z) for the elements of Mob(D)
we have a canonical form of A = ±R1 HR2 , where R1 and R2 are rotation matrices and
c
0
H=
.
0 c−1
4
JAIRO BOCHI
Proposition 6. Given A ∈ SL(2, R) and v ∈ R2 with |v| = 1, let ξ be the point of ∂D
associated to the direction of v under the homeomorphism φ. Then
−1/2
|Av| = |fA0 (ξ)|
.
A heuristic argument using infinitesimals goes as follows. A maps the unit circle onto
an ellipse, preserving area. The slice of the unit disk corresponding to the element of angle
2
dθ around v, whose area is 2dθ, is mapped to a slice of the ellipse with area 2 |Av| dϕ.
−2
Therefore, |fA0 (ξ)| = dϕ/dθ = |Av| . A formal proof follows.
az + b
a b
be the corresponding element of Mob(H).
Proof. Let A =
and g(z) =
c d
cz + d
Then fA = h ◦ g ◦ h−1 . Consider a vector w = (x, 1) ∈ R2 and let ξ ∈ ∂D be associated
to the direction of w. We may consider the map h in (3) as an automorphism of the
Riemann sphere C ∪ {∞} = CP1 = S2 . Then h is an isometry relative to the metric
defined on the sphere by
2 |dz|
.
kdzkz = 2
|z| + 1
So, considering fA and g also as maps on the sphere,
kD(fA )ξ k = Dhg(x) ◦ Dgx ◦ D(h−1 )ξ = kDgx k .
Now, kD(fA )ξ k = |fA0 (ξ)| and
kDgx k = |g 0 (x)|
k·kg(x)
k·kx
2
=
|ad − bc| |x| + 1
2
2
|cx + d| |g(x)| + 1
2
=
2
|x| + 1
2
2
=
|w|
2.
|ax + b| + |cx + d|
|Aw|
Normalizing the vector w we obtain the relation in the statement.
=
Given A ∈ SL(2, R) which is not a rotation, let s(A) ∈ RP1 the direction that is most
contracted and u(A) ∈ RP1 the direction the direction that is most expanded by A. For
simplicity, we shall write also u(A), s(A) for φ(u(A)), φ(s(A)).
f −1 (0)
Corollary 7. We have s(A) = A−1 and u(A) = −s(A).
f (0)
A
Proof. Putting propositions 5 and 6 together we get
ξ − f −1 (0)
A
0
−1/2
|Av| = |fA (ξ)|
=q
.
2
1 − |fA (0)|
Thus, the most contracted direction corresponds to the point ξ ∈ ∂D closest to fA−1 (0),
which is fA−1 (0)/|fA−1 (0)|. Analogously for the most expanded direction.
Corollary 7 also shows that the directions s(A), u(A) ∈ RP1 are orthogonal.
q
1+|fA (0)|
Proposition 8. kAk = 1−|f
= exp( 21 ρ(fA (0), 0)) for all A ∈ SL(2, R).
A (0)|
Proof. If A is a rotation then kAk = 1 and fA (0) = 0, and the equalities are clear.
Otherwise, if A is not a rotation, proposition 6 gives
u(A) − f −1 (0)
−1/2
A
kAk = |A u(A)| = |fA0 (u(A))|
= q
.
2
1 − |fA (0)|
OSELEDETS
5
Using proposition 5 and noting that |fA−1 (0)| = |fA (0)|, we conclude that
p
1 + fA−1 (0)
1 + |fA (0)|
=p
kAk = q
2
1 − |fA (0)|
1 − |fA (0)|
as claimed.
Lyapunov exponents for Mob(D)-Cocycles. An SL(2, R)-cocycle naturally induces a
PSL(2, R)-cocycle and, conversely, a PSL(2, R)-cocycle can always be lifted to a SL(2, R)cocycle 2. By definition, the Lyapunov exponents of a PSL(2, R)-cocycle are those of any
such lift, the choice being irrelevant. Using PSL(2, R) = Mob(D) we get a notion of
Lyapunov exponent for Mob(D)-cocycles. This can be made explicit using proposition 8:
writing fx = fFx , and fxn = fFxn = fT n−1 x ◦ · · · ◦ fx for each n, we find
lim
n→±∞
1
1
log kFxn k = lim
ρ(fxn (0), 0).
n→±∞
n
2n
In other words, the Lyapunov exponents correspond to the rates of growth of the hyperbolic
distance from fxn (0) to the origin. So, in this context the theorem of Oseledets becomes:
Theorem 9. Let T : (X, µ) → (X, µ) be an invertible ergodic transformation and f :
X → Mob(D) be a measurable map such that
Z
(4)
ρ(fx (0), 0) dµ(x) < ∞.
X
Then there exists λ ≥ 0 such that
1
ρ(fxn (0), 0) = 2λ
n→∞ n
(5)
lim
for µ-almost every x ∈ X.
Furthermore, if λ > 0 there are measurable maps ws , wu : X → ∂D such that
1
2λ if z = ws (x)
lim
log |(fxn )0 (z)| =
n→+∞ n
−2λ if z ∈ D with z 6= ws (x),
1
log |(fxn )0 (z)| =
n→−∞ n
lim
−2λ if z = wu (x)
2λ if z ∈ D with z 6= wu (x).
If λ = 0 then
1
log |(fxn )0 (z)| = 0 for all z ∈ D.
n→±∞ n
lim
Remark.
(1) In view of (1), the relation (5) is equivalent to
lim
n→∞
1
log (1 − |fxn (0)|) = 2λ
n
for µ-almost every x.
(2) The contents of (4) and (5) do not change if we replace the origin by any other
point in the open disk.
(3) If the system is not ergodic λ is a function of x but, otherwise, the theorem
remains valid.
2This is one place where it is simpler to work on a trivial bundle. For general vector bundles, this lift
can be done locally, and all that follows is easily adapted.
6
JAIRO BOCHI
Proof of the Oseledets theorem in dimension 2
We shall use Kingman’s sub-additive ergodic theorem in the following form:
Theorem 10 ([Kin68]). Assume T is ergodic. If (ϕn )n=1,2,... is a sequence of functions
m
such that ϕ+
for all m, n ≥ 1. Then n1 ϕn
1 is integrable and ϕm+n ≤ ϕm + ϕn ◦ T
converges a.e. to some c ∈ R ∪ {−∞}.
Define ϕn (x) = ρ(fxn (0), 0). Then ϕm+n ≤ ϕm + ϕn ◦ T m , by the triangle inequality.
Using theorem 10 we get that n1 ϕn converges µ-almost everywhere to a constant 2λ. Since
ϕn ≥ 0, λ ≥ 0. This proves (5).
Define wn (x) = (fxn )−1 (0) for every integer n. Notice that, by the invariance of the
hyperbolic metric, ρ(wn (x), 0) = ρ(fxn (0), 0). Using (1) we get, for a.e. x,
(6)
lim
n→+∞
1
log(1 − |wn (x)|) = −2λ
n
Suppose that λ > 0. Then the distance from wn to the origin goes to infinity, which
means that wn converges to the boundary of D as n → ∞.
Lemma 11. We have lim sup
n→+∞
1
log |wn+1 (x) − wn (x)| ≤ −2λ for µ-a.e. x.
n
Proof. Since the hyperbolic metric is invariant under automorphisms,
(7)
ρ(wn+1 , wn ) = ρ((fxn )−1 ◦ (fT n x )−1 (0), (fxn )−1 (0)) =
= ρ((fT n x )−1 (0), 0) = ρ(fT n x (0), 0).
The idea of the proof is that if ρ(fT n x (0), 0) is not too big, that is, if wn+1 and wn are not
too far away from each other in terms of the hyperbolic metric, then the Euclidian distance
between wn+1 and wn will have to be exponentially small, since wn → ∂D exponentially
fast. Write bn (x) = fT n (x) (0), for simplicity. For a.e. x, we have
(8)
1
ρ(bn (x), 0) → 0 .
n
This follows from proposition 3 applied to the function ϕ(x) = ρ(fx (0), 0), which, by
assumption (4), is integrable. Fix x in the full measure set where (6) and (8) hold. In
view of (1)–(2) the equality (7) implies
|wn+1 − wn |
= |bn |
|1 − wn wn+1 |
or, equivalently,
2
|wn+1 − wn | = |bn | |1 − wn wn+1 | = |bn | 1 − |wn | + wn (wn − wn+1 )
2
≤ |bn | 1 − |wn | + |wn | |wn+1 − wn | .
That is,
2
|wn+1 − wn | ≤
|bn |(1 − |wn | )
.
1 − |bn | |wn |
Since |wn | < 1 and |bn | < 1, the last inequality implies
2
|wn+1 − wn | <
1 − |wn |
2 (1 − |wn |)
<
.
1 − |bn |
1 − |bn |
OSELEDETS
The condition (8) is equivalent to
inequality above, we conclude
lim sup
n→+∞
1
n
7
log(1 − |bn |) → 0. Combining this with (6) and the
1
log |wn+1 − wn | ≤ −2λ.
n
The lemma implies that wn (x) = (fxn )−1 (0) is a Cauchy sequence for almost every x.
Let ws (x) ∈ ∂D be the limit. Notice that, by corollary 7,
wn (x)
= lim s(fxn ).
n→∞
n→∞ |wn (x)|
ws (x) = lim wn (x) = lim
n→∞
Let us show how to compute the growth rate of log |(fxn )0 (z)| for z ∈ ∂D. By proposition 5,
and using |fxn (0)| = |(fxn )−1 (0)|, we have
2
|(fxn )0 (z)| =
1 − |fxn (0)|
|z −
2
(fxn )−1 (0)|
= (1 + |wn |)
1 − |wn |
.
|z − wn |2
Using (6) we deduce
(9)
lim
n→+∞
1
1
log |(fxn )0 (z)| = −2λ − 2 lim
log |z − wn | .
n→+∞ n
n
For all z 6= ws (x) this gives that the limit is −2λ.
Now consider the case z = ws (x). Since |wn −ws | ≥ 1−|wn |, we have lim inf n1 log |ws −
wn | ≥ −2λ. On the other hand, take 0 < ε < 2λ. By lemma 11, we have |wj+1 − wj | ≤
e(−2λ+ε)j if j is large enough. Hence
|ws − wn | ≤
∞
X
|wj+1 − wj | ≤
j=n
e(−2λ+ε)j
,
1 − e−2λ+ε
and so lim sup n1 log |ws − wn | ≤ −2λ + ε. This proves that lim n1 log |ws − wn | = −2λ.
Substituting in (9), we get lim n1 log |(fxn )0 (z)| = 2λ.
Now we do the corresponding calculation for z ∈ D. By the invariance of the hyperbolic
metric under fxn , we have
2
(10)
|(fxn )0 (z)| =
1 − |fxn (z)|
2
1 − |z|
=
1 − |wn |2
.
1 − |z|2
It follows, using (6), that lim n1 log |(fxn )0 (z)| = 2λ.
The statements about wu follow by symmetry, considering the inverse cocycle.
At last, we consider the case λ = 0. If z ∈ ∂D, then using (9) and 1−|wn | ≤ |z−wn | ≤ 2,
we get lim n1 log |(fxn )0 (z)| = 0. If z ∈ D then we simply use (10).
The proof of theorem 9 is complete.
Remark. Using proposition 3, it is easy to prove that if λ > 0 then
1
log |wu (T n x) − ws (T n x)| = 0.
n→±∞ n
lim
This corresponds to the fact that the angles between Oseledets directions are sub-exponential.
8
JAIRO BOCHI
Higher dimensions. The Oseledets theorem for the disk D extends directly to the hyperbolic ball Bn of dimension n ≥ 2. The formula (2) can be generalized as follows. Given
two points in Bn consider the disk that contains both and the origin. Since this disk is
isometric to D, we may use (2) to compute the distance between the points. In the case
n = 3 there is an isomorphism between the groups Mob(B3 ) and PSL(2, C), induced
by identifying ∂B3 = CP1 . It is an extension of the isomorphism between Mob(D) and
PSL(2, R) that we used above, and all the relations between real matrices and automorphisms of the disk that we proved extend to complex matrices and automorphisms of the
ball. We ignore whether there is a nice matrix representation of the groups Mob(Bn ) for
n ≥ 4.
References
[Kin68] J. Kingman. The ergodic theorem of subadditive stochastic processes. J. Royal Statist. Soc.,
30:499–510, 1968.
[Mañ87] R. Mañé. Ergodic theory and differentiable dynamics. Springer Verlag, 1987.
[Nic89] P. J. Nichols. The Ergodic Theory of Discrete Groups. Cambridge Univ. Press, 1989.
[Ose68] V. I. Oseledets. A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc., 19:197–231, 1968.
[You95] L.-S. Young. Ergodic theory of differentiable dynamical systems. In Real and Complex Dynamical Systems, volume NATO ASI Series, C-464, pages 293–336. Kluwer Acad. Publ., 1995.